First, we write an equation to represent that the fencing lengths add up to 568 feet. we call the side of the fence that has three segments of its length x and the side with only two segments y. We write 3x + 2y = 568. We also know that the area of the rectangle is equal to xy, so area = xy. We put y in terms of x using our first equation and find that y = (568 - 3x)/2. We plug this into our area equation and find that area = (568x - 3x^2)/2. To find the maximum we set the derivative equal to 0 and end up with 0 = 284 - 3x. We solve for x and get 94 and 2/3. We then put that into our first equation to find y = 142. So the dimensions that maximize the area are 94 2/3 x 142.